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in the figure, a vector \\(\vec{a}\\) with a magnitude of 15.0 m is directed at angle \\(\theta = 79^\circ\\) counterclockwise from the +x axis. what are the components (a) \\(a_x\\) and (b) \\(a_y\\) of the vector? a second coordinate system is inclined by angle \\(\theta = 19^\circ\\) with respect to the first. what are the components (c) \\(a_x\\) and (d) \\(a_y\\) in this primed coordinate system?
(a) number
units
(b) number
units
(c) number
units
(d) number
units
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Explanation:

Step1: Find $a_x$ (x-component)

$a_x = |\vec{a}| \cos\theta$
$a_x = 15.0 \cos(79^\circ)$
$a_x \approx 15.0 \times 0.1908 = 2.86$

Step2: Find $a_y$ (y-component)

$a_y = |\vec{a}| \sin\theta$
$a_y = 15.0 \sin(79^\circ)$
$a_y \approx 15.0 \times 0.9816 = 14.72$

Step3: Find angle for primed x-axis

The angle of $\vec{a}$ relative to the $x'$-axis is $\theta - \theta' = 79^\circ - 19^\circ = 60^\circ$
$a'_x = |\vec{a}| \cos(\theta - \theta')$
$a'_x = 15.0 \cos(60^\circ)$
$a'_x = 15.0 \times 0.5 = 7.50$

Step4: Find $a'_y$ (primed y-component)

$a'_y = |\vec{a}| \sin(\theta - \theta')$
$a'_y = 15.0 \sin(60^\circ)$
$a'_y = 15.0 \times \frac{\sqrt{3}}{2} \approx 12.99$

Answer:

(a) 2.86 Units: m
(b) 14.72 Units: m
(c) 7.50 Units: m
(d) 12.99 Units: m